The accidental crossing of energy levels is studied for a number of exactly solvable PT-symmetric potentials in one spatial dimension. This phenomenon occurs when the potential possesses two series of bound-state levels discriminated by the q=± quasi-parity quantum number and a potential parameter is tuned to specific values. In contrast with the coalescing of two such real-energy levels with the same n quantum number and continuing as a complex conjugate pair, corresponding to the breakdown of PT symmetry, accidental crossing occurs for energy levels with different n and q. In this case the energy eigenvalues become degenerate, and the corresponding wave functions become linearly dependent. It is shown that besides the known examples, the PT-symmetric harmonic oscillator, Coulomb and Scarf II potentials, this phenomenon occurs for any member of the Natanzon potential class for which the q quantum number can be defined. Two such potentials are discussed as concrete examples: the PT-symmetric generalized Ginocchio potential and a four-parameter subset of the Natanzon potential class. These potentials have been described in detail previously, however, the accidental crossing of their energy eigenvalues has not been noticed then.
ASJC Scopus subject areas
- Physics and Astronomy(all)